
Steve Ritter '85
I think if you ask most high school students what mathematicians do, they'd
say "solve really hard equations." Despite being a pretty good math student
in high school and having some extracurricular interests in fields like
topology, I probably would have given the same answer when I was in high
school. Like many students, I didn't realize that the things I found most
interesting about math were also the things that mathematicians did. The
skillsbased academic curricula provide the means to do interesting work in
mathematics, but skills aren't what the field is about.
Tom Banchoff was largely responsible for helping me realize that the
mathematical concepts that I found interesting were also worthy of being
taken seriously. This was the essence of his Mathematical Way of Thinking
course and also his contribution when, as part of a computer graphics
course, Kevin Pickhardt and I developed a program to graph three and
fourdimensional surfaces, given an implicit equation. Kevin and I
implemented the program, but Tom provided the fundamental insight that made
the algorithm work. He taught us to treat the surface as a real object
situated in space. This was a simple, but crucial, change in perspective
for us.
Kevin and I had tested the system on spheres and the occasional torus, but
Tom wanted to put the software to work right away. He immediately entered
the equation for a surface mentioned (but not illustrated) in a paper he was
reading (well, it wasnÕt quite immediate — we first needed to expand the
size of the text field to accommodate the equation). The three of us
anxiously stared at the screen as the surface started to render. Panic
initially set in when Tom muttered, "wait, thatÕs not it," but as the full
image appeared and was rotated to a better viewing angle, we were satisfied
and relieved that the program worked.
I was a Cognitive Science major at Brown and earned a Ph.D. in Cognitive
Psychology at Carnegie Mellon University in 1992. In retrospect, perhaps it
is not surprising that I was able to combine my interest in cognition with
my interests in math. As a postdoc, I started developing cognitive models
of how students thought about mathematics and incorporated them into
"intelligent tutoring systems" that help students learn math. These systems
became successful enough to lead to a business (Carnegie Learning, where I
now work). Our "Cognitive Tutors" are now being used as primary curricula
in over 1500 schools.
The focus, of course, is on mathematical concepts. In an indirect way, I
think these courses are helping to transmit the kind of excitement and
vitality that Tom Banchoff helped me see in mathematics to a new generation
of students.
— Steve Ritter October 2003 
